Methods 41 (2007) 238–247
www.elsevier.com/locate/ymeth
Mathematical modeling as a tool for investigating cell
cycle control networks
Jill C. Sible *, John J. Tyson
Department of Biological Sciences, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0406, USA
Accepted 11 July 2006
Abstract
Although not a traditional experimental ‘‘method,’’ mathematical modeling can provide a powerful approach for investigating complex cell signaling networks, such as those that regulate the eukaryotic cell division cycle. We describe here one modeling approach based
on expressing the rates of biochemical reactions in terms of nonlinear ordinary differential equations. We discuss the steps and challenges
in assigning numerical values to model parameters and the importance of experimental testing of a mathematical model. We illustrate
this approach throughout with the simple and well-characterized example of mitotic cell cycles in frog egg extracts. To facilitate new
modeling efforts, we describe several publicly available modeling environments, each with a collection of integrated programs for mathematical modeling. This review is intended to justify the place of mathematical modeling as a standard method for studying molecular
regulatory networks and to guide the non-expert to initiate modeling projects in order to gain a systems-level perspective for complex
control systems.
! 2006 Elsevier Inc. All rights reserved.
Keywords: Cell cycle; Computational biology; Frog egg extracts; Mathematical modeling; Ordinary differential equations; Parameter estimation;
Systems biology; Xenopus laevis
1. Introduction
Because a universal mechanism controlling DNA synthesis, mitosis, and cell division underlies the growth,
development, and reproduction of all eukaryotes, an
understanding of this molecular regulatory system is one
of the most important goals of modern cell biology. As
the complex network of cell cycle controls is uncovered,
it becomes increasingly difficult to make reliable predictions about how modification of one component affects
the system as a whole. However, such predictions are needed if we are to identify the host of mutations contributing
to cancer or find within the molecular network novel
targets for therapeutic intervention. Mathematical models
provide powerful tools for managing the complexity of
the cell cycle control system and of other signaling
*
Corresponding author. Fax: +1 540 231 9307.
E-mail address: [email protected] (J.C. Sible).
1046-2023/$ - see front matter ! 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.ymeth.2006.08.003
networks. Models organize a large body of experimental
data, describe the fundamental behaviors of the system as
a whole, bridge gaps where experimental data are missing,
and drive hypothesis-building for the next round of experimentation. The value of mathematical modeling in
describing and predicting the behavior of complex systems
has been well established in fields such as chemical engineering and meteorology, but its power has been underappreciated until recently in molecular cell biology.
Although mathematical models can be built to describe
any signaling network, application of modeling tools to cell
cycle regulation is particularly well suited and timely. First,
the data in this field are vast, both providing a large body
of information to build comprehensive models and creating
the need for a tool to understand how these data fit together. Second, cell cycle signaling networks are modular,
allowing models to be constructed in parts and then assembled and reassembled in various ways. Furthermore, many
models of the network are comparable between different
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J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247
tively simple, many predictions of the model have been validated experimentally, and this module can be adapted to
more complex cell cycles both in Xenopus and other
eukaryotes.
organisms (e.g., budding yeast and mammals) so that it is
feasible to make relatively small changes to a model
describing one particular system in order to apply it to
another. Thus, each new model need not be constructed
from scratch. Third, a reasonable amount of quantitative
or semi-quantitative information can be extracted from
the literature, facilitating the early phases of model building. By modifying established protocols (described in the
accompanying articles), additional quantitative data can
be generated to improve parameter estimation and experimental validation of models. Finally, despite the wealth of
detailed information on cell cycle molecules and their specific interactions, we lack a systems-level perspective of this
complex control network. Modeling can provide this perspective by helping to identify underlying regulatory principles. Where a specific experimental detail is missing,
modeling can serve as bridge, enabling progress in building
a systems-level view, and guiding the design and execution
of future experiments.
Although the term ‘‘mathematical modeling’’ encompasses a wide range of computational approaches applicable to cell cycle studies, we focus this review on one branch
of modeling: The construction of ordinary differential
equations (ODEs) to describe protein interaction networks
that regulate the cell cycle. This approach has a strong
track record of yielding accurate, predictive and testable
models [1–7], and in recent years, has been integrated well
with experimental methodologies [8–12]. We will illustrate
each step of the modeling process with an example: The
core signaling network that controls entry into and exit
from mitosis in frog (Xenopus laevis) egg extracts by regulating the activity of the mitotic cyclin-dependent kinase,
Cdk1. We chose this example because the network is rela-
amino acids
P
P
Cdk1
Cyclin
2. Approach
2.1. Construction of a wiring diagram
The first step in the modeling process is to organize the
known interactions of the relevant molecules into a concept
map or wiring diagram. A simple diagram representing the
core machinery regulating mitotic transitions in Xenopus
cell-free egg extracts is depicted in Fig. 1. The diagram
depicts each biochemical entity with a separate icon. For
example, the phosphorylated form of Wee1 is represented
by a modified version of the icon for unphosphorylated
Wee1. Solid arrows indicate chemical transitions between
states, and dotted arrows represent a modulating signal
on a biochemical reaction (often the catalytic influence of
an enzyme). Because no universal conventions for building
these diagrams exist, all symbols must be defined explicitly
and used consistently. Hopefully, contributors to this field
will adopt a common symbolism in the near future,
facilitating the exchange of information between wiring
diagrams and their affiliated mathematical models.
Despite the wealth of information regarding cell cycle
signaling networks, gaps in knowledge may exist such
that portions of the diagram cannot be wired with certainty. It is neither necessary nor advised to postpone
the modeling process indefinitely until all molecular
details of the wiring diagram are understood. Rather,
the gaps in the wiring diagram can be filled with place-
Cyclin
P
Cdk1
Cdc25
P
Cdc25
Cdk1
Cyclin
Wee1
IE
IE
AP
P
Wee1
OFF
APC
C
ON
Cdk1
P
degraded
cyclin
Fig. 1. Wiring diagram of the core mitotic cell cycle engine in Xenopus cell-free egg extracts. Central to the diagram is the active cyclin–Cdk1 complex (also
referred to as MPF), with activating phosphorylation on Thr 161 indicated by the green ‘P’. Cyclin enters the system by de novo synthesis and then
combines with Cdk1. The phosphorylation on Thr 161 is rapid and therefore not represented in the diagram or the mathematical equations. Active Cdk1
can be phosphorylated on Thr 14 and Tyr 15 to form inactive preMPF. The double arrows indicate the reversibility of this process. The inhibitory
phosphates are represented by the red ‘P’; and the influences of the relevant kinase (Wee1) and phosphatase (Cdc25C) are indicated by the dotted arrows.
MPF itself phosphorylates Cdc25 and Wee1, positively and negatively affecting their activities, respectively. These feedback loops are easily appreciated
from the wiring diagram. MPF also participates in a negative feedback loop whereby it phosphorylates a component of the APC, which subsequently
directs the polyubiquitination of cyclin, tagging it for degradation by the proteasome. Like cyclin synthesis, cyclin degradation is represented by a single
unidirectional arrow because the process is irreversible. Although not indicated in the diagram, cyclin degradation by the APC applies equally to free
cyclin monomers and to preMPF. IE indicates an intermediate component (see text).
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holders. These placeholders can be generic ones for which
parameters will be selected empirically to model the behavior of the system without representing the molecular mechanism in detail. This approach allows the modeling effort to
continue in order to discover properties of the system as a
whole as well as specific features of better-characterized
portions of the diagram. Alternatively, a more specific
placeholder can be created that makes some untested
assumptions about the molecular mechanism. These
assumptions in the wiring diagram will be tested in simulations of the model to determine whether they are consistent
with known behaviors of the system. Eventually, the
assumptions that work in silico must be tested experimentally when appropriate methodologies and reagents become
available. Ideally, portions of the wiring diagram that are
hypothetical should be clearly delineated from those that
are grounded in experimentation.
Our example, a model of the cell cycle machinery regulating mitosis in frog egg extracts, demonstrates the usefulness of placeholders. When the wiring diagram depicted in
Fig. 1 was first constructed [2], little information was
known about the mechanisms regulating exit from mitosis.
Experimental data suggested that MPF triggered the degradation of cyclin [13] and a previous mathematical model
indicated that a time delay in this negative feedback loop
could create sustained oscillations [3]. Based on this
information, the wiring diagram was constructed with an
intermediate component (called IE) that was phosphorylated by MPF, and then activated the APC, triggering
the polyubiquitination and subsequent degradation of
cyclin [2]. Several years later as the details of the ubiquitin–proteasome pathway governing cyclin degradation
were discovered, IE was replaced by Fizzy/Cdc20 and components of the APC [14–16] but the fundamental features
of the model remained the same.
2.2. Writing the ordinary differential equations
Once the model is represented diagrammatically, the
biochemical relationships need to be converted to the language of mathematics in the form of nonlinear ODEs. A
rate equation is written for each biochemical entity whose
concentration changes over time. The right side of the
equation includes a positive term for every arrow that
directs toward that entity and a negative term for every
arrow that directs away from the entity. The kinetics selected should be appropriate for the type of biochemical transition represented. In general, mass-action kinetics are
appropriate for most biochemical reactions except when
[substrate] ! [enzyme], in which case Michaelis–Menten
kinetics should be applied.
In our example of the core mitotic machinery, the rate
equation for the concentration of cyclin monomers is based
on mass-action kinetics:
d
½Cyclin# ¼ k 1 % k 2 ½Cyclin# % k 3 ½Cyclin#½Cdk#
dt
where k1 is the rate constant for cyclin synthesis, k2 is a
function that describes the activity of the APC in promoting cyclin degradation, and k3 is the rate constant for association of cyclin monomers and Cdk monomers to form
MPF dimers. These dimers are phosphorylated on threonine 161 to make active MPF. Because this reaction is fast
and essentially irreversible, we are justified in neglecting the
T161-unphosphorylated forms of MPF.
The rate equation for [MPF] is likewise based on massaction kinetics:
d
½MPF# ¼ k 3 ½Cyclin#½Cdk# % k 2 ½MPF#
dt
% k wee ½MPF# þ k 25 ½preMPF#
A similar equation pertains to [preMPF]. [Cdk], the concentration of Cdk monomers, is governed not by an
ODE but rather by a conservation condition:
½Total Cdk# ¼ ½Cdk# þ ½MPF# þ ½preMPF# ¼ constant:
We assume that [Total Cdk] = 100 nM in a typical frog egg
extract.
The full set of differential equations for the model of Mphase control is shown in Fig. 2. This early version of the
model uses Michaelis–Menten kinetics to describe the rate
changes of the regulatory enzymes, Cdc25 and Wee1 [2].
Because of the positive feedback between Cdc25 and
MPF and between Wee1 and MPF, the assumption that
[enzyme] ' [substrate] may be invalid. In more recent iterations of the model, these equations have been rewritten
with mass-action kinetics, and parameter sets that reproduce the fundamental behaviors of the system have been
identified (Dravid and Tyson, unpublished data).
2.3. Parameterizing the model
The next step in the modeling process is the most challenging. A preliminary set of parameters (rate constants
for each reaction) must be selected. The challenges arise
because for many models, few if any of these parameters
have been measured directly. Therefore, the initial set of
parameters must be selected based on semi-quantitative
data, indirect measurements and typically, a fair amount
of guesswork. In an iterative process, each set of parameters must be ‘‘run’’ through the model (described below)
and modified to bring the output of the model into better
and better agreement with observed behaviors of the
system.
Even though few direct measurements of rate constants
or concentrations of cell-cycle regulatory proteins have
been published, valuable information for parameter estimation can be mined from the literature, and a thorough
effort to do so is recommended before one undertakes the
task of measuring parameters de novo. A reasonable body
of experimental literature has informed the model depicted
in Figs. 1 and 2, although the initial parameter set was
almost purely guesswork, based on rough qualitative
features of mitotic control in frog eggs and extracts [2].
J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247
241
Fig. 2. Mathematical model of the cell cycle control network in frog egg extracts [2]. ODEs describe the rate of change in concentration of every nonconstant species in the wiring diagram in Fig. 1.
Although enzyme rate constants are rarely measured or
reported directly, reasonable estimates can be derived from
time course studies, which abound in the cell cycle
literature. For example, the experimental data of Kumagai
and Dunphy [17], reproduced in Fig. 3, gives us hints about
the rates of phosphorylation of cyclin-Cdk1 by Wee1 in its
more active form (V 00wee ) and its less active form (V 0wee ).
Lanes d–h and i–m show that Cdk1 is rapidly phosphorylated by Wee1 in an interphase extract, when Wee1 is
expected to be in its active form. That rapid activation is
observed even in the presence of aphidicolin (‘‘+APC;’’
Fig. 3. Experimental data used to estimate the value of V 00wee in the NovakTyson model of the cell cycle in frog egg extracts. Data are from Fig. 3C of
Kumagai and Dunphy, 1995 (17). ‘‘Cdc2’’ refers to Cdk1 (lower band is
unphosphorylated on Wee1 sites and upper band is phosphorylated on
Wee1 sites). APC = aphidicolin, an inhibitor of DNA replication and
CAFF = caffeine, an inhibitor of cell cycle checkpoint kinases.
not to be confused with the Anaphase Promoting Complex), an inhibitor of DNA replication. Because Cdk1
appears to be completely converted to the tyrosine phosphorylated form within 2 minutes, V 00wee must be 1 min%1
or larger. On the other hand, in an M-phase arrested
extract (lanes a–c), tyrosine phosphorylation of Cdk1 by
less active Wee1 is much slower, say V 0wee ( 0:01min%1 .
Kumagai and Dunphy (1995) [17] also show that the rate
of tyrosine phosphorylation is 2.5 times slower when cycloheximide-treated extracts were supplemented with cyclin B
monomers rather than with preformed cyclin B-Cdk1
dimers (their Fig. 3A and C), allowing Marlovits et al.
(1998) [43] to estimate that k3 ( 0.005 nM%1 min%1.
Although only rough estimates, these values for V 0wee ,
V 00wee , and k3 provide good starting points for more sophisticated parameter optimization procedures. Table 1 provides the ODE files with the Marlovits’ estimates of the
rate constants for the ODEs in Fig. 2.
At present, mining the literature to inform parameter
estimation remains a laborious task. The body of literature
is enormous and even the most carefully constructed
database search algorithms will not identify all of the articles that might contain experiments relevant for parameter
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J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247
Table 1
Input file for WinPP to reproduce Fig. 4
# A model of MPF regulation in frog egg extracts
# after (Novak & Tyson, J. Cell Sci., 1993)
# Warning: WinPP and XPP are case insensitive.
# Differential Equations (Figure 2)
dCyclin/dt = k1- k2*Cyclin - k3*Cyclin*Cdk
dMPF/dt = k3*cyclin*Cdk- k2*MPF - kwee*MPF + k25*preMPF
dpreMPF/dt = kwee*MPF- k25*preMPF - k2*preMPF
dCdc25P/dt = F(TotCdc25-Cdc25P,MPF,ka,KKa) - F(Cdc25P,PPase,kb,KKb)
dWee1P/dt = F(TotWee1-Wee1P,MPF,ke,KKe) - F(Wee1P,PPase,kf,KKf)
dIEP/dt = F(TotIE-IEP,MPF,kg,KKg) - F(IEP,PPase,kh,KKh)
dAPC/dt = F(TotAPC-APC,IEP,kc,KKc) - F(APC,PPase,kd,KKd)
# Conservation of Cdk subunits
Cdk = TotCdk - MPF - preMPF
# Rate functions embodying the regulatory signals
k25 = V25'*(TotCdc25-Cdc25P) + V25"*Cdc25P
kwee = Vwee'*Wee1P + Vwee"*(TotWee1-Wee1P)
k2 = V2'*(TotAPC-APC) + V2"*APC
# Michaelis-Menten Rate Law
F(S,E,k,Km) = k*E*S/(Km+S)
# Parameter values (from Marlovits et al. [43])
par k1=1, V2'=.005, V2"=.25, k3=.005
par V25'=.017, V25"=.17, Vwee'=.01, Vwee"=1
par ka=.02, KKa=.1, kb=.1, KKb=1
par kc=.13, KKc=.01, kd=.13, KKd=1
par ke=.02, KKe=.1, kf=.1, KKf=1
par kg=.02, KKg=.01, kh=.15, KKh=.01
par TotCdk=100
# These parameters are arbitrarily set to 1.
par TotCdc25=1, TotWee1=1, TotIE=1, TotAPC=1, PPase=1
# Initial conditions
init Cyclin=0, MPF=0, preMPF=0, Cdc25P=0, Wee1P=0, IEP=1, APC=1
# Define "auxiliary" variables for plotting purposes
aux TotCyclin=cyclin+MPF+preMPF
aux Wee1=10*(TotWee1-Wee1P)
# Simulator settings
@ meth=stiff, Total=200, Bounds=1000
# Plotter settings
@ xhi=200, ylo=0, yhi=25
@ nplot=2, yp2=MPF
done
J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247
A
simulations (described below), parameters can be twiddled
with efficiency so that the modeler can develop a sense of
which variables under which conditions are sensitive to
which parameters. Such simulation will help design experiments that will yield the most quantitative ‘‘bang’’ for
one’s funding ‘‘buck.’’ If during this twiddling phase, no
reasonable parameter set can be developed to approximate
the behavior of the system, then the wiring diagram may
need to be reconsidered, and experiments carried out to test
fundamental qualitative assumptions of the model rather
than generating specific quantitative data.
25
period = 55 min
concentration (nM)
20
total cyclin
15
10
MPF
5
0
pre-MPF
0
50
100
150
200
2.4. Running simulations of the model
time (min)
B
80
total cyclin = 80 nM
70
70
[MPF] (nM)
60
60
50
50
40
40
30
30
20
20
10
15
0
0
5
10
15
20
25
30
243
35
40
time (min)
Fig. 4. Simulations of the mathematical model of the cell cycle in frog egg
extracts describe known and previously unobserved behaviors of the
control system. (A) Simulation of the changes in cyclin, preMPF and MPF
levels over time in a cycling frog egg extract [2]. Computed from the ‘ode’
file in Table 1. The simulations are consistent with a body of experimental
data. (B) Simulations of the lag time into mitosis after addition of various
fixed conc entrations of nondegradable cyclin B. Computed from the ‘ode’
file in Table 1, with k 1 ¼ V 02 ¼ V 002 ¼ 0. The lengthening lag times at
progressively lower concentrations of cyclin B was not validated experimentally until 2003, when sufficiently small increments in cyclin B were
used for the experiments [8].
estimation. As modeling efforts become more widely utilized and organized, links between specific components of
the model (parameters or details of the wiring diagram)
and the data upon which they are based (a specific figure
or page in a research article) need to be built. Meanwhile,
even a limited amount of quantitative information can be
useful in constructing the initial set of parameters,
constraining portions of the model and allowing the governing ODEs to be numerically simulated to see how close
or far the model lies from describing observed behaviors.
Once the process of mining the literature has been
exhausted, then the remaining parameters must be estimated by other means. Even in cases where the majority of the
parameter values are unknown or where it would be feasible to measure parameter values directly, we recommend
that the initial parameter set be derived with a minimal
investment in new experimentation. Through computer
With an initial parameter set in hand, the next step is
to ‘‘run’’ the model, that is, to use software that solves
ODEs numerically in order to simulate the behavior of
the system over time. WinPP (for Windows) and XPP
(for Unix) are two simulation programs freely available
from Dr. G. Bard Ermentrout, University of Pittsburgh.
Go to http://www.math.pitt.edu/)phase for links to software downloads and tutorials. The basic output of a
simulation is a graph of the concentration of each component over time.
To run a simulation one needs, in addition to differential
equations (Fig. 2) and rate constants (Table 1), ‘‘initial conditions’’ for all the variables (i.e., concentrations of each
variable at t = 0). Initial conditions can usually be estimated readily from the experimental protocol of a particular
experiment. For example, a ‘‘cycling’’ extract is prepared
by activating the APC, which degrades most of the cyclin
in the extract, so it is reasonable to set [IEP] = [APC] = 1
and [cyclin] = [MPF] = [preMPF] = 0 at t = 0. Furthermore, we assume that [Cdc25P] = [Wee1P] = 0 at t = 0,
because MPF is inactive. These common sense considerations fix reasonable initial conditions for a simulation of
the temporal evolution of a cycling extract.
The ODEs, parameter values, and initial conditions are
communicated to WinPP in a text file called an ‘-.ode’ file
(see Table 1). By opening this -.ode file in WinPP.exe and
clicking ‘Run’ ‘Go,’ one will generate Fig. 4A, which simulates the temporal evolution of cyclin B, preMPF and
active MPF in a cycling frog-egg extract. A comparison
of the simulation to experimental observations shows that
the model successfully reproduces the basic features of the
egg extract: Cyclin accumulates during interphase then is
rapidly degraded at mitosis, and a time lag exists in which
tyrosine-phosphorylated cyclin:Cdk dimers form, then
these dimers are converted to the active form of MPF.
The simulation represents the behavior of the cell cycle
engine in an unperturbed extract. One could also vary
parameters to represent experiments in which one or more
components of the system is altered. For example, in a classic set of experiments by Solomon et al. [18], synthesis of
endogenous cyclin was blocked by addition of cycloheximide, and extracts were supplemented with fixed amounts
of recombinant, non-degradable cyclin B (Dcyclin B).
J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247
2.5. Parameter optimization
Initial simulations of the model, based on the original
estimates of parameter values, may or may not approximate the behavior of the system to the modeler’s satisfaction. The more qualitative data available to build the
wiring diagram and quantitative data available to constrain
the parameter values, the closer the simulations are likely
to be to the biological system. However, rarely have all rate
constants been measured, and estimates of initial conditions typically rely on semi-quantitative and relative data
(such as that obtained by Western blotting). Therefore,
most parameter sets will need to be optimized to bring
the model and the biological system into better quantitative
agreement. For relatively simple models, such as our example here, consisting of 7 ODEs and 24 kinetic constants, it
may be possible to derive a satisfying parameter set ‘‘by
hand,’’ varying one parameter at a time and determining
how well simulations match known behaviors of the system. However, this approach is inefficient and tedious,
requires considerable advance knowledge of the system in
order to make a good ‘‘starting guess’’ of the parameters,
and is virtually impossible for more complex models consisting of many variables and many more parameters.
A collection of global parameter optimization tools is
publicly available in the Gepasi modeling platform (discussed below) [19]. The choice of which tool to apply
remains somewhat empirical [19,20] and largely the realm
of the theoretician.
In our example of the cell cycle engine in frog egg
extracts, a deterministic global optimizer (called VTDIRECT) followed by local optimization (using ODRPACK, an implementation of the Levenberg–Marquardt
method) was applied to derive an optimized set of rate
constants [21]. Overall, simulations using the optimized
parameter set give a better fit to the experimental data
than simulations using the Marlovits parameter values
(Table 1).
A
MPF activity
As done originally by Novak and Tyson (1993) [2],, we can
simulate this experiment by setting k 1 ¼ V 02 ¼ V 002 ¼ 0 and
setting the initial concentration of cyclin to a fixed value.
By varying the initial concentration of cyclin (Fig. 4B),
we can draw two conclusions from the model. First of
all, the graph of active MPF over time indicates that the
initial cyclin concentration must exceed a threshold
(between 15 and 20 nM in Fig. 4B) in order to activate
MPF. This prediction agrees well with the experimental
data [18]. The simulation also suggests that as cyclin concentration approaches this activation threshold from larger
values, the lag time to activate MPF becomes longer and
longer. This prediction was first made by Novak and Tyson
in 1993. Although no published studies showed evidence of
this ‘‘critical slowing down,’’ subsequent experiments confirmed this prediction [8]. This example illustrates that
mathematical models of the cell cycle can both be informed
by and can inform experimental studies.
Ti
cyclin level
Ta
B 25
(V’wee = V’’ wee = 0)
period = 47 min
20
concentration (nM)
244
15
total cyclin
MPF
10
5
0
0
50
100
150
200
time (min)
Fig. 5. Theoretical predictions from Novak and Tyson [2]. (A) Hysteresis
drives mitotic transitions in frog egg extracts. The simulation indicates
that the concentration of cyclin required to drive high MPF activity (and
thereby enter mitosis) is higher than the concentration required to
maintain high MPF activity (and thereby stay in mitosis). For the
Marlovits parameter set, Table 1, Ti = 8 nM and Ta = 17 nM. The
theoretical prediction of two distinct thresholds was confirmed experimentally [8, 9]. (B) MPF oscillations are faster and smaller if Cdk1 is not
phosphorylatable by Wee1. Computed from the ‘ode’ file in Table 1, with
V 0wee ¼ V 00wee ¼ 0. This prediction was confirmed experimentally by
Pomerening et al. [11].
2.6. Experimental testing of the model
Once parameter optimization is used to bring a mathematical model into reasonable quantitative and qualitative
agreement with existing experimental data, the next step is
to test predictions made by the model experimentally. The
rationale for doing so is twofold: the predictive ability of a
model is a rigorous test of its accuracy, and more importantly, a major reason for building a model in the first place
is to make novel predictions that inform subsequent rounds
of experimentation. Virtually any experiment can be simulated in silico by varying parameter values in logical ways:
(e.g., decreasing an enzyme’s rate constant to represent the
effect of a specific inhibitor, setting the rates of synthesis of
all proteins to zero in the presence of cycloheximide, setting
the rate of synthesis of a specific protein to zero for a gene
J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247
deletion mutant, or setting the initial concentration of protein to zero for an immunodepletion). Conditions that produce the most interesting and measurable cell behaviors are
good candidates to test experimentally. Other considerations are whether or not the experiment is feasible (e.g.,
depleting a Cdk without removing associated cyclins could
be problematic) and whether a particular experiment will
help define ambiguous parts of the model (e.g., resolve
uncertainties in the wiring diagram or constrain a parameter value).
The interplay between modeling and experimentation is
at its most valuable when a model can lead to novel, nonintuitive predictions about the behavior of the system. For
example, Novak and Tyson’s prediction that a hysteresis
loop governs entry into and exit from mitosis in frog egg
extracts (Fig. 5A), was tested and validated experimentally
[8,9]. Using a modified protocol of Solomon et al. [18], in
which cyclin levels were controlled in cycloheximide-treated extracts by the addition of exogenous nondegradable
cyclin B, a variety of experiments indicated that higher levels of cyclin are required to enter mitosis than to stay in
mitosis. Another key prediction of the Novak–Tyson model is that, if the control of MPF activity by Wee1-dependent
phosphorylation of Cdk1 is eliminated, then MPF activity
continues to oscillate but at a higher frequency and smaller
amplitude (compare Figs. 4A and 5B). This prediction was
validated experimentally in a series of clever experiments in
which Wee1- and Cdc25-control loops were eliminated by
adding a nonphosphorylatable Cdk1 (called the Cdc2AF
mutant) and inhibiting endogenous Cdk1 [11]. These elegant yet straightforward experiments were inspired by systems-level predictions derived from mathematical models
rather than the published experimental studies that preceded them.
Because the frog egg extract is biochemically tractable,
experimental testing of the model was performed by biochemical manipulations, such as altering the concentration
of cyclin in the system. Genetic approaches can be used to
test both quantitative and qualitative predictions of the
models that describe cell cycle networks in genetically tractable organisms. For example, a comprehensive mathematical model of the cell cycle control network in
Saccharomyces cerevisiae describes the behavior of more
than 100 genetic mutants [4,22]. Rigorous tests of the model have been performed by creating new mutants predicted
by the model to give the most informative phenotypes [23]
and by challenging the model to predict phenotypes of yet
unpublished mutants [24]. Complete details of the model
are available at an easily navigable web site: http://
mpf.biol.vt.edu/research/budding_yeast_model/pp/.
3. Modeling tools and environments
The development of modeling ‘‘environments’’ that contain suites of tools necessary for model building, simulations, data fitting, and data management is an
advancement that will benefit both experts and novices
245
building models of molecular regulatory networks. The
modeling environments described here are publicly available, relatively user friendly, and operate on personal computers running Windows, Macintosh, and/or Linux
operating systems. Furthermore, each environment contains tools to translate models into the Systems Biology
Markup Language (SBML), a grammar that is becoming
widely adopted by the biochemical network modeling community to exchange models [25].
3.1. Gepasi (http://www.gepasi.org)
Gepasi, the first package for modeling biochemical
networks, was originally released in 1993 [26] and has been
enhanced by regular additions and revisions [27]. The
current version (3.3) runs on Windows and Linux operating systems. Gepasi supports modeling of time course data
in an environment that includes a graphical user interface
(GUI) for model building, tools for simulations, and a
collection of parameter estimation algorithms. Gepasi
allows the definition of spatial compartments and includes
a tool called MEG that allows modeling of spatially
heterogeneous systems such as cell cultures or tissues [28].
The environment also includes a number of tutorials to
guide novice users through key aspects of the modeling
environment. Although the Gepasi 3.3 is reported to be
the last iteration, COPASI (complex pathway simulator),
a new modeling environment based on Gepasi, is being
developed. Test versions of COPASI are available at
http://www.copasi.org. COPASI is designed to be more
user-friendly, compatible with the latest versions of SMBL,
and functional in Windows, Macintosh and Linux operating systems.
3.2. Virtual cell (http://www.vcell.org)
Virtual Cell, a modeling environment created by the
National Resource for Cell Analysis and Modeling,
may be the most user-friendly for the non-expert [29].
The current version, 4.0, can be run as a Java application or applet link on Windows, Macintosh OSX 10.1
and higher, and Linux operating systems. User support
is available by e-mail. Programs for building models,
running simulations, and managing documents are
included, with plans for parameter estimation tools in
Version 4.2. Virtual Cell places particular emphasis on
modeling spatially distributed systems, providing tools
to formulate and simulate partial differential equations
(PDEs), which are better suited to represent spatiotemporal dynamics in cell signaling networks [30]. Virtual
Cell 4.1, available as a beta version, allows representation of two- and three-dimensional membrane diffusion
and visualization of three-dimensional surfaces. Virtual
Cell utilizes its own language, VCML, which can be
translated to SBML. A host of known incompatibilities
between the two languages is documented on the Virtual
Cell website.
246
J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247
3.3. Systems biology workbench (http://sbw.kgi.edu)
The Systems Biology Workbench (SBW) provides a
suite of tools generically suited for modeling biochemical
networks [31]. The current version 2.5.0 operates in
Windows, Linux and Macintosh OSX. SBW provides a
GUI for model building (JDesigner) and a simulation tool
(Jarnac) [32]. Most importantly, SBW enables modeling
applications written in diverse languages to communicate
and operate together. The SBW group has worked for
interoperability with the BioSPICE modeling projects,
described below. SBW requires some background in
dynamical systems and computer science to utilize the full
suite of programs, but the basic operation of JDesigner/
Jarnac is designed to be user-friendly.
3.4. JigCell (http://jigcell.biol.vt.edu)
JigCell is a modeling environment for Windows being
developed as part of the larger DARPA BioSPICE effort
[33,34]. Although still a work-in-progress, JigCell includes
a convenient spreadsheet-style Model Builder, a Run Manager for organizing related parameter sets and running
multiple simulations [35], a Comparator for comparing
simulations to experimental data [36], and a Parameter
Estimator [21,37]. Although JigCell is suited for modeling
any biochemical network, it was developed with a particular emphasis on modeling the cell cycle. A model of the cell
cycles of frog egg extracts (described above) and a model of
the more complex budding yeast cell cycle have been used
as test cases for developing the various components of JigCell, and these models are available on the website. JigCell
is designed for interoperability with other modeling environments under the BioSPICE umbrella, and thus, is
compatible with SBW as well.
ing will take its place in the molecular cell biologist’s
toolkit alongside SDS–PAGE and PCR!
5. For further study
This review was written for the non-expert. We recommend as a first foray into modeling, that newcomers familiarize themselves with the example provided here and/or
tutorial models found on the websites described in Section
3. For those interested in delving deeper into the theory
and practice of modeling complex biochemical networks,
we suggest literatures [38–42].
Acknowledgments
We thank the NIH (GM59688 to J.C.S) and DARPA/
ARFL (F30602-01-2-0572 to J.J.T) for funding. Thanks
to Dr. Brian Wroble for critical reading of this
manuscript.
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